A136485 Number of unit square lattice cells enclosed by origin centered circle of diameter n.

0, 0, 4, 4, 12, 16, 24, 32, 52, 60, 76, 88, 112, 120, 148, 164, 192, 216, 256, 276, 308, 332, 376, 392, 440, 476, 524, 556, 608, 648, 688, 732, 796, 832, 904, 936, 1012, 1052, 1124, 1176, 1232, 1288, 1372, 1428, 1508, 1560, 1648, 1696, 1788, 1860, 1952, 2016

Offset

1,3

Comments

a(n) is the number of complete squares that fit inside the circle with diameter n, drawn on squared paper.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

a(n) = 4 * Sum_{k=1..floor(n/2)} floor(sqrt((n/2)^2 - k^2)).

a(n) = 4 * A136483(n).

a(n) = 2 * A136513(n).

Lim_{n -> oo} a(n)/(n^2) -> Pi/4 (A003881).

a(n) = [x^(n^2)] (theta_3(x^4) - 1)^2 / (1 - x). - Ilya Gutkovskiy, Nov 24 2021

Example

a(3) = 4 because a circle centered at the origin and of radius 3/2 encloses (-1,-1), (-1,1), (1,-1), (1,1).

Mathematica

Table[4*Sum[Floor[Sqrt[(n/2)^2 - k^2]], {k, Floor[n/2]}], {n, 100}]

Prog

(Magma)
A136485:= func< n | n le 1 select 0 else 4*(&+[Floor(Sqrt((n/2)^2-j^2)): j in [1..Floor(n/2)]]) >;
[A136485(n): n in [1..100]]; // G. C. Greubel, Jul 29 2023
(SageMath)
def A136485(n): return 4*sum(floor(sqrt((n/2)^2-k^2)) for k in range(1, (n//2)+1))
[A136485(n) for n in range(1, 101)] # G. C. Greubel, Jul 29 2023

Crossrefs

Cf. A003881, A136483, A136513.

Alternating merge of A119677 of A136485.

Sequence in context: A282503 A323189 A121189 * A157617 A053415 A303315

Adjacent sequences: A136482 A136483 A136484 * A136486 A136487 A136488

Keyword

easy,nonn

Author

Glenn C. Foster (gfoster(AT)uiuc.edu), Jan 02 2008

Status

approved